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Axiom ax-mulass 7395
Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7355. Proofs should normally use mulass 7420 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 7295 . . . 4 class
31, 2wcel 1436 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1436 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 1436 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 922 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 7302 . . . . 5 class ·
101, 4, 9co 5615 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 5615 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 5615 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 5615 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1287 . 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
158, 14wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Colors of variables: wff set class
This axiom is referenced by:  mulass  7420
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